Numerical study of the wrinkling of a stretched thin sheet

A thin sheet clamped at opposite ends and stretched develops wrinkles parallel to the direction of the applied tensile strain due to the hindered Poisson lateral contraction at the clamps. To study this phenomenon, a variational model recently proposed by Puntel, Deseri and Fried is adopted. The relevant energy functional includes bending and membranal contributions and is minimized subject to a constraint on the area of the mid-surface of the sheet. A fourth order partial-differential equation is henceforth obtained and numerically implemented using B-splines. Predictions are obtained concerning the number of wrinkles, critical applied stretches, and scaling relationships for wrinkle amplitude and wavelength. Both a linearized version of the boundary-value problem based on the small-slope approximation and a fully nonlinear one are considered: their results are found to be in good agreement for the whole range of applied stretches taken into account. Comparisons with previous analytical results by Puntel, Deseri and Fried, who used different boundary conditions and an Ansatz on the deflection function are also provided. The numerical results substantially confirm the validity of the analytical predictions. The present work provides then an alternative numerical method for the study of wrinkling in thin sheets and supports the use of analytical and semi-analytical solutions as viable options for specific geometries. Though further investigation, particularly experimental, is still needed, extensive comparisons of the results with other studies available in the literature provide confirmation for the scaling laws and signal that predicted values of the critical stretches may only be accurate for higher length-to-width aspect ratios.